# Is there an abstract characterization of freeness in terms of additive unitary cocycles?

This question is very closely related to my other question here.

Let $\Gamma$ be a countable discrete group and $\pi:\Gamma \rightarrow \mathcal{H}$ be a unitary representation of $\Gamma$. A map $b:\Gamma \rightarrow \mathcal{H}$ is an additive cocycle if $$b(gh)=\pi(g)b(h)+b(g)$$ for all $g,h\in G$.

If we consider the collection of all $(b,\pi)$ where $\pi$ is a unitary representation of $\Gamma$ and $b$ is a cocycle as above, can this collection "see" if $\Gamma$ is a free group?

Again, I'm being deliberately vague here regarding the word "collection". Roughly, what I am asking is whether including these cocycles in my previous question would yield a structure that can detect freeness of the underlying group.

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Maybe this is what you are looking for: If $\Gamma$ is free on the set $X$, then for any unitary representation $\pi \colon \Gamma \to U(H)$ any map $c \colon X \to H$ extends uniquely to a $1$-cocycle on $\Gamma$. Moreover, this happens if and only if $\Gamma$ is free. In other words, the linear space of 1-cocycles for $\pi$ on $\Gamma$ is precisely the space of $H$-valued functions on $X$.
This observation can be used to show that the first $\ell^2$-Betti number of $\Gamma$ is $|X|-1$. (Minus one, because you have to substract the inner 1-cocycles.) The first $\ell^2$-Betti number of $\Gamma$ precisely counts how many 1-cocycles with values in the left-regular representation $\lambda$ exist.
One does not even have to know this for all unitary representations $\pi$, it is enough to know it for the left-regular representation $\lambda$. Indeed, a group on $n$ generators is free if and only if its first $\ell^2$-Betti number equals $n-1$. This is sharp, as their are $n$-generated groups which are not free but the first $\ell^2$-Betti number exceed $n-1 - \varepsilon$, for example $\Gamma = \langle a,b \mid a^k \rangle$ for $k$ high enough. More interestingly, Denis Osin has constructed $n$-generated torsion groups with first $\ell^2$-Betti number higher than $n-1 - \varepsilon$.